CONTEMPORARY MATHEMATICS

432

In the Tradition of Ahlfors-Bers/ IV

Ahlfors-Bers Colloquium May 19-22, 2005

University of Michigan Ann Arbor, Michigan

Dick Canary Jane Gilman

Juha Heinonen Howard Masur

Editors

In the Tradition of Ahlfors-Bers, IV

http://dx.doi.org/10.1090/conm/432

CoNTEMPORARY MATHEMATICS

432

In the Tradition of Ahlfors-Bersl IV

Ahlfors-Bers Colloquium May 19-22, 2005

University of Michigan Ann Arbor, Michigan

Dick Canary Jane Gilman

Juha Heinonen Howard Masur

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor

George Andrews Andreas Blass Abel Klein

2000 Mathematics Subject Classification. Primary 14H15, 20H10, 28A75, 30F40, 30C62, 32G15, 54E40, 57M50.

Library of Congress Cataloging-in-Publication Data Ahlfors-Bers Colloquium (4th : 2005 : Ann Arbor, Michigan)

In the tradition of Ahlfors-Bers, IV : the Ahlfors-Bers Colloquium, May 19-22, 2005, Ann Arbor, Michigan / Dick Canary ... [et a!.], editors.

p. em. -(Contemporary mathematics, ISSN 0271-4132 ; 432) Includes bibliographical references. ISBN-13: 978-0-8218-4227-0 (alk. paper) ISBN-10: 0-8218-4227-7 (alk. paper) 1. Functions-Congresses. 2. Riemann surfaces-Congresses. 3. Mappings (Mathematics)-

Congresses. I. Canary, Richard Douglas. II. Title.

QA331.A462 2005 515--dc22 2007060747

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10987654321 12 11 10 09 08 07

Contents

Preface vii

Plenary Talks IX

Workshops and Workshop Speakers xi

Colloquium Participants xiii

Uniformly exponential growth and mapping class groups of surfaces J. W. ANDERSON, J. ARAMOYONA, and K. J. SCHACKLETON 1

An A1 weight not comparable with any quasiconformal Jacobian CHRISTOPHER J. BISHOP 7

Curvature, stretchiness, and dynamics MOON DUCHIN 19

Some special loci in the Siegel space of genus two CLIFFORD J. EARLE 31

Another approach to the automorphism theorem for Teichmiiller spaces EGE FUJIKAWA 39

Energy of twisted harmonic maps of Riemann surfaces WILLIAM M. GOLDMAN and RICHARD A. WENTWORTH 45

Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis PETER HASTO, ZAIR IBRAGIMOV, DAVID MINDA, SAMINATHAN PONNUSAMY, and 8WADESH 8AHOO 63

From left earthquakes to right JuN Hu 75

Small eigenvalues and maximal laminations on complete surfaces of negative curvature

CHRISTOPHER M. JUDGE 93

A generalized hyperbolic metric for plane domains LINDA KEEN and NIKOLA LAKIC 107

Subgroups of mapping class groups from the geometrical viewpoint RICHARD P. KENT IV and CHRISTOPHER J. LEININGER 119

Determinants of Laplacians, quasifuchsian spaces, and holomorphic extensions of Laplacians

YOUNG-HEON KIM 143

v

vi CONTENTS

Hyperbolic and quasisymmetric structure of hyperspace LEONID V. KOVALEV and JEREMY T. TYSON

A classification of the modular transformations of infinite dimensional Teichmiiller spaces

KATSUHIKO MATSUZAKI

Random hyperbolic surfaces and measured laminations MARYAM MIRZAKHANI

Extensions of holomorphic motions to quasiconformal motions SUDEB MITRA

Analyst's Traveling Salesman Theorems. A Survey RAANAN SHUL

Energy of harmonic maps and Gardiner's formula RICHARD A. WENTWORTH

151

167

179

199

209

221

Preface

The Ahlfors-Bers Colloquia commemorate the mathematical legacy of Lars Ahlfors and Lipman Bers. The core of this legacy lies in the fields of geometric function theory, Teichmiiller theory, hyperbolic manifolds, and partial differential equations. However, their work has impacted and created interactions with many other fields, such as algebraic geometry, mathematical physics, dynamics, geomet-ric group theory, number theory and topology. These triannual colloquia serve as a venue to disseminate the work since the previous colloquium to the wider mathematical community and bring the key participants together to ponder future directions in the field.

The third Ahlfors-Bers Colloquium was held in Ann Arbor, Michigan in May 2005 and was attended by nearly 150 mathematicians from eight countries. This was a particularly exciting time for the colloquium to take place as many old and well known problems had been resolved since the last colloquium. Here are some high-lights. In the theory of hyperbolic 3-manifolds, Ahlfors Measure Conjecture, the Bers-Sullivan-Thurston Density Conjecture, Marden's Tameness Conjecture and Thurston's Ending Lamination Conjecture had all been resolved, and new phenom-ena in the deformation theory of Kleinian groups had been revealed. The study of mapping class groups had been re-invigorated by techniques from geometric group theory and Teichmiiller theory. In the theory of meromorphic functions, the spher-ical Bloch constant had been determined. Versions of the uniformization theorem had been found that were purely metric in formulation, with applications to geo-metric group theory. The conformal invariance of percolation had been established.

In addition to the plenary talks, there were three pre-colloquium workshops that focused on early career mathematicians. The workshops were Conformal Analysis and Geometric Function Theory organized by Lukas Geyer and Sergiy Merenkov, Deformation Theory of Hyperbolic 3-manifolds organized by Dick Canary and Jane Gilman, and Weil-Petersson Geometry of Teichmiiller space organized by Scott Wolpert and Mike Wolf. The list of plenary and workshop speakers and their titles is given below.

The first Colloquium in this series was the Bers Colloquium (1995). This was followed by the first Ahlfors-Bers colloquium (1998) and the second Ahlfors-Bers colloquium (2001). Proceeding of these three colloquia have been published in the AMS Contemporary Mathematics Series, Lipa's Legacy, volume 211, In the Tradition of Ahlfors-Bers, volume 256, In the Tradition of Ahlfors-Bers, III, volume 335, respectively. One important part of the Ahlfors-Bers legacy is to cast a wide net that includes old and young, new and established researchers. We hope that with this volume the legacy of Ahlfors and Bers will continue to have an impact.

vii

viii 1. PREFACE

AB05 was supported by the NSF and the University of Michigan. The orga-nizing committee consisted of Bill A bikoff, Dick Canary (Chair), Alex Eremenko, Jane Gilman, Juha Heinonen, Howie Masur, Yair Minsky, Michael Wolf, and Scott Wolpert. The editors of this volume and the organizing committee thank the mem-bers of the local organizing committee, Dick Canary, Lukas Geyer, Juha Heinonen, and Sergiy Merenkov, for making the colloquium a success.

Dick Canary Jane Gilman

J uha Heinonen Howard Masur

Plenary Speakers with Titles

Agol, Ian Tameness of negatively curved 3-manifolds

Astala, Kari Complex analysis, impedance tomography and quasiconformal mappings

Bishop, Chris Conformal mapping in linear time

Bonk, Mario Conformal Analysis on Fractals

Calegari, Danny Shrinkwrapping and the taming of hyperbolic 3-manifolds

Kerckhoff, Steve Boundary value problems in deformation theory

Minsky, Yair Ends, Laminations and Families of Surfaces

Mirzakhani, Maryam Random hyperbolic surfaces and measured laminations

Rees, Mary The Ending Laminations Theorem Direct from Teichmuller Geodesics.

Souto, Juan Volume of hyperbolic manifolds and distances of Heegaard splittings

Traizet, Martin Riemann surfaces with nodes and minimal surfaces without symmetries.

ix

Workshops and Speakers

Conformal Analysis and Geometric Function Theory organizers: Lukas Geyer and Sergiy Merenkov Deformation Theory of Hyperbolic 3-manifolds

organizers: Dick Canary and Jane Gilman Weil-Petersson Geometry of Teichmiiller space

organizers: Scott Wolpert and Mike Wolfe

Jeremy Best Constants for weighted Moser-Trudinger inequalities on sub-Riemannian spaces

Brock, Jeffrey Chains of flats, minimal irrational laminations, and divergent geodesics in the Weil-Petersson metric

Choi, Young Lines of minima in Teichmuller space

DeMarco, Laura Polynomial dynamics and trees

Duchin, Moon Variation of Teichmuller geodesics

Dumas, David Grafting, the Schwarzian derivative, and complex projective structures on surfaces

Evans, Richard Bounded balls in convex cores

Hasti::i, Peter Isometries of some Mobius and similarity invariant metrics

xi

Huang, Zheng The Canonical metric on a Riemann surface and its induced metric on the moduli

Ito, Kentaro 3-dimensional extension of the Maskit slice for once-punctured tori

Kim, Young-Heon Determinants of Laplacians and their holomorphic extensions

Kovalev, Leonid Conformal dimension of metric spaces

Leininger, Chris Shadows of mapping class groups: capturing convex co-compactness.

Margalit, Dan Weil-Petersson isometries via the pants complex

Marshall, Tim The smallest covolume Kleinian group

Mcintyre, Andrew An explicit Belavin-Knizhnik formula

Milley, Peter Cusp diagrams and small hyperbolic 3-manifolds

xii WORKSHOPS, ORGANIZERS, SPEAKERS AND TITLES

Miyachi, Hideki Moduli of continuity of Cannon-Thurston maps

Namazi, Hossein Heegaard splittings and hyperbolic geometry

Pilgrim, Kevin Coarse conformal dynamics

Purcell, Jessica Geometry of complicated knots and links

Raft, Kasra A combinatorial model for the Teichmuller metric

Roeder, Roland Combinitorial aspects of Andreev's classification of Hyperbolic Polyhedra.

Saric, Dragomir Decorated Teichmiiller space of the punctured solenoid

Schul, Raanan A characterization of subsets of finite length curves in Hilbert space and the analyst's TSP

Storm, Peter A group theoretic application of the Marden conjecture

Sun, Xiaofeng Good metrics on the moduli space of Riemann surfaces

Tyson, Jeremy Best constants for weighted Moser-Tridinger inequalities on sub-Riemannian spaces

Uriarte-Thero, Ignacio On Marcinkiewicz integrals and harmonic measure

Colloquium Participants

Bill Abikoff University of Connecticut

Ilesanmi A de boye University of Michigan

Ian Agol University of Illinois at Chicago

Jim Anderson University of Southampton

Kari Astala University of Helsinki

Christopher Atkinson University of Illinois at Chicago

David Barrett University of Michigan

Ara Basmajian University of Oklahoma

Thomas Bieske University of South Florida

Chris Bishop SUNY Stony Brook

Mario Bonk University of Michigan

Joshua Bowman Cornell University Math Department

Jeffrey Brock Brown University

Ken Bromberg U ofUT

Danny Calegari California Institute of Technology

Dick Canary University of Michigan

Christopher Cashen University of Illinois, Chicago

Young-Eun Choi University of California, Davis

Suhyoung Choi KAIST

Adam Coffman IU Purdue-Fort Wayne

Serban Costea University of Michigan

David Crombecque University of Southern California

Jason DeBlois University of Texas-Austin

Laura DeMarco University of Chicago

Casey Douglas Rice

Moon Duchin University of Chicago

David Dumas Rice University

Nathan Dunfield Cal tech

Anirban Dutta SUNY at Stony Brook

Clifford Earle Cornell University

xiii

xiv PARTICIPANTS

Richard Evans U. of Auckland

Erlend Fornaess Wold

Ege Fujikawa Tokyo Institute of Technology

Vaibhav Gadre Cal tech

Frederick Gardiner Brooklyn College, CUNY

Frederick Gehring U of Michigan

Lukas Geyer University of Michiga

Jane Gilman Rutgers-Newark

Jasun Gong University of Michigan

Hrant Hakobyan SUNY, Stony Brook University

Peter Hasto Norwegian University of Science and Technolgy

Juha Heinonen U. Michigan

Chris Hiatt University of Southern California

Pat Hooper Yale and SUNY, Stony Brook

Jun Hu CUNY, Brooklyn College and Graduate Center

Zheng Huang University of Michigan

Zair Ibragimov University of Cincinnati

Nikolaev Igor University of Calgary

Elmas Irmak University of Michigan

Kentaro Ito Graduate School of Mathematics, Nagoya University

Prabhu Janakiraman University of Illinois, Urbana-Champaign

Lizhen Ji Univ of Michigan

Chris Judge Indiana University

Jason Karcher University of Illinois, Chicago

Robert Kaufman University of Illinois

Linda Keen CUNY, Lehman College and Graduate Center

Richard Kent University of Texas at Austin

Steve Kerckhoff Stanford University

Young-Heon Kim Northwestern University

Soomin Kim Rice University

Youngju Kim University of Oklahoma

Leonid Kovalev Washington University in St. Louis

Enrico Le Donne Yale University

Christopher Leininger Columbia U.

Anna Lenzhen University of Illinois at Chicago

Dean Leonardi University of Illinois at Chcago

PARTICIPANTS

John Mackay U of Michigan

John Maki University of Illinois at Urbana-Champaign

Johanna Mangahas University of Michigan

Al Marden University of Minnesota

Dan Margalit University of Utah

Tim Marshall North Dakota State University

Howard Masur UIC

Katsuhiko Matsuzaki Ochanomizu University

Andrew Mcintyre Ohio State University

Jean McKemie St. Edward's University

Ben McReynolds University of Texas

Ivana Medos Columbia University

Sergiy Merenkov University of Michigan

Peter Milley University of California at Riverside

Yair Minsky Yale University

Maryam Mirzakhani Princeton University /Clay

Sudeb Mitra CUNY, Queens College

Hideki Miyachi Tokyo Denki University

Bryan Mosher University of Michigan

Roman Muchnik University of Chicago

Gregory Muller Cornell University

Kei Nakamura UC Davis

Hossein N amazi Yale & SUNY Stony Brook

Yusuke Okuyama Kanazawa University & Helsinki University

J ani Onninen University of Michigan

Barbara Paier Rice University

Seungkyu Park usc Kevin Pilgrim Indiana University

Jessica Purcell University of Texas at Austin

Kasra Rafi University of Connecticut

Kai Rajala University of Michigan

Mary Rees University of Liverpool

Roland Roeder Cornell University

Dragomir Saric SUNY Stony Brook

Saul Schleimer Rutgers

Ben Schmidt University of Michigan

Raanan Schul Yale University

Peter Scott U Michigan

XV

xvi

Ali Shaker University of Illinois at Chicago

Marie Snipes University of Michigan

Juan Souto Laboratoire Emile Picard, CNRS, Toulouse

Ralf Spatzier University of Michigan

Peter Storm Stanford

Xiaofeng Sun Harvard University

Perry Susskind

PARTICIPANTS

Matthias Weber Indiana University,

Richard Wentworth Johns Hopkins University

Adam Weyhaupt Indiana University- Bloomington

Kevin Wildrick University of Michigan

Michael Wolf Rice University

Scott Wolpert University of Maryland

Jang-Mei Wu University of Illinois

Connecticut College/Williams College Shanshuang Yang Emory University

Jing Tao UIC

Edward Taylor Wesleyan University

Josh Thompson University of Utah

Martin Traizet Universite de Tours, France

Jeremy Tyson University of Illinois at Urbana-Champaign

Ignacio Uriarte-Thero Helsinki University

Adrian Vajiac Chapman University

Sergey Vasilyev University of Chicago

Liz Vivas University of Michigan

Fang Wang Northwestern Unveristy

Biao Wang Cornell University

Shenglan Yuan GC-CUNY

Wei Zhu Rice University

Eric Zupunski University of Michigan

Titles in This Series

432 Dick Canary, Jane Gilman, Juha Heinonen, and Howard Masur, Editors, In the tradition of Ahlfors-Bers, IV, 2007

431 Michael Batanin, Alexei Davydov, Michael Johnson, Stephen Lack, and Amnon Neeman, Editors, Categories in algebra, geometry and mathematical physics, 2007

430 Idris Assani, Editor, Ergodic theory and related fields, 2007 429 Gui-Qiang Chen, Elton Hsu, and Mark Pinsky, Editors, Stochastic analysis and

partial differential equations, 2007 428 Estela A. Gavosto, Marianne K. Korten, Charles N. Moore, and Rodolfo H.

Torres, Editors, Harmonic analysis, partial differential equations, and related topics, 2007 427 Anastasios Mallios and Marina Haralampidou, Editors, Topological algebras and

applications, 2007 426 Fabio Ancona, Irena Lasiecka, Walter Littman, and Roberto Triggiani, Editors,

Control methods in POE-dynamical systems, 2007 425 Su Gao, Steve Jackson, and Yi Zhang, Editors, Advances in Logic, 2007 424 V. I. Burenko, T. Iwaniec, and S. K. Vodopyanov, Editors, Analysis and geometry

in their interaction, 2007 423 Christos A. Athanasiadis, Victor V. Batyrev, Dimitrios I. Dais, Martin Henk,

and Francisco Santos, Editors, Algebraic and geometric combinatorics, 2007 422 JongHae Keum and Shigeyuki Kondo, Editors, Algebraic Geometry, 2007 421 Benjamin Fine, Anthony M. Gaglione, and Dennis Spellman, Editors,

Combinatorial group theory, discrete groups, and number theory, 2007 420 William Chin, James Osterburg, and Declan Quinn, Editors, Groups, rings and

algebras, 2006 419 Dinh V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and Its

applications, 2006 418 Lothar Gerritzen, Dorian Goldfeld, Martin Kreuzer, Gerhard Rosenberger,

and Vladimir Shpilrain, Editors, Algebraic methods in cryptography, 2006 417 Vadim B. Kuznetsov and Siddhartha Sahi, Editors, Jack, Hall-Littlewood and

Macdonald polynomials, 2006 416 Toshitake Kohno and Masanori Morishita, Editors, Primes and knots, 2006 415 Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment,

Editors, Quantum graphs and their applications, 2006 414 Deguang Han, Palle E. T. Jorgensen, and David Royal Larson, Editors, Operator

theory, operator algebras, and applications, 2006 413 Georgia M. Benkart, Jens C. Jantzen, Zongzhu Lin, Daniel K. Nakano,

and Brian J. Parshall, Editors, Representations of algebraic groups, quantum groups and Lie algebras, 2006

412 Nikolai Chernov, Yulia Karpeshina, Ian W. Knowles, Roger T. Lewis, and Rudi Weikard, Editors, Recent advances in differential equations and mathematical physics, 2006

411 J. Marshall Ash and Roger L. Jones, Editors, Harmonic analysis: Calder6n-Zygmund and beyond, 2006

410 Abba Gumel, Carlos Castillo-Chavez, Ronald E. Mickens, and Dominic P. Clemence, Editors, Mathematical studies on human disease dynamics: Emerging paradigms and challenges, 2006

409 Juan Luis Vazquez, Xavier Cabre, and Jose Antonio Carrillo, Editors, Recent trends in partial differential equations, 2006

408 Habib Ammari and Hyeonbae Kang, Editors, Inverse problems, multi-scale analysis and effective medium theory, 2006

TITLES IN THIS SERIES

407 Alejandro Adem, Jesus Gonzalez, and Guillermo Pastor, Editors, Recent developments in algebraic topology, 2006

406 Jose A. de Ia Pefia and Raymundo Bautista, Editors, Trends in representation theory of algebras and related topics, 2006

405 Andrew Markoe and Eric Todd Quinto, Editors, Integral geometry and tomography, 2006

404 Alexander Borichev, Hakan Hedenmalm, and Kehe Zhu, Editors, Bergman spaces and related topics in complex analysis, 2006

403 Tyler J. Jarvis, Takashi Kimura, and Arkady Vaintrob, Editors, Gromov-Witten theory of spin curves and orbifolds, 2006

402 Zvi Arad, Mariagrazia Bianchi, Wolfgang Herfort, Patrizia Longobardi, Mercede Maj, and Carlo Scoppola, Editors, Ischia group theory 2004, 2006

401 Katrin Becker, Melanie Becker, Aaron Bertram, PaulS. Green, and Benjamin McKay, Editors, Snowbird lectures on string geometry, 2006

400 Shiferaw Berhanu, Hua Chen, Jorge Hounie, Xiaojun Huang, Sheng-Li Tan, and Stephen S.-T. Yau, Editors, Recent progress on some problems in several complex variables and partial differential equations, 2006

399 Dominique Arlettaz and Kathryn Hess, Editors, An Alpine anthology of homotopy theory, 2006

398 Jay Jorgenson and Lynne Walling, Editors, The ubiquitous heat kernel, 2006 397 Jose M. Muii.oz Porras, Sorin Popescu, and Rub{ E. Rodriguez, Editors, The

geometry of Riemann surfaces and Abelian varieties, 2006 396 Robert L. Devaney and Linda Keen, Editors, Complex dynamics: Twenty-five years

after the appearance of the Mandelbrot set, 2006 395 Gary R. Jensen and Steven G. Krantz, Editors, 150 Years of Mathematics at

Washington University in St. Louis, 2006 394 Rostislav Grigorchuk, Michael Mihalik, Mark Sapir, and Zoran Sunik, Editors,

Topological and asymptotic aspects of group theory, 2006 393 Alec L. Matheson, Michael I. Stessin, and Richard M. Timoney, Editors, Recent

advances in operator-related function theory, 2006 392 Stephen Berman, Brian Parshall, Leonard Scott, and Weiqiang Wang, Editors,

Infinite-dimensional aspects of representation theory and applications, 2005 391 Jiirgen Fuchs, Jouko Mickelsson, Grigori Rozenblioum, Alexander Stolin,

and Anders Westerberg, Editors, Noncommutative geometry and representation theory in mathematical physics, 2005

390 Sudhir Ghorpade, Hema Srinivasan, and Jugal Verma, Editors, Commutative algebra and algebraic geometry, 2005

389 James Eells, Etienne Ghys, Mikhail Lyubich, Jacob Palis, and Jose Seade, Editors, Geometry and dynamics, 2005

388 Ravi Vakil, Editor, Snowbird lectures in algebraic geometry, 2005 387 Michael Entov, Yehuda Pinchover, and Michah Sageev, Editors, Geometry,

spectral theory, groups, and dynamics, 2005 386 Yasuyuki Kachi, S. B. Mulay, and Pavlos Tzermias, Editors, Recent progress in

arithmetic and algebraic geometry, 2005 385 Sergiy Kolyada, Yuri Manin, and Thomas Ward, Editors, Algebraic and topological

dynamics, 2005

For a complete list of titles in this series, visit the AMS Bookstore at www.arns.org/bookstore/.

The Ahlfors-Bers Colloquia commemorate the mathematical legacy of Lars Ahlfors and Lipman Bers. The core of this legacy lies in the fields of geometric function theory, Teichmiiller theory, hyperbolic manifolds, and partial differential equations. However, the work of Ahlfors and Bers has impacted and created interactions with many other fields, such as algebraic geometry, mathematical physics, dynamics, geometric group theory, number theory, and topology. The triannual Ahlford-Bers colloquia serve as a venue to disseminate the relevant work to the wider mathematical community and bring the key participants together to ponder future directions in the field. The present volume includes a wide range of articles in the fields central to this legacy. The majority of articles present new results, but there are expository articles as well.

ISBN 978-0-8218-4227-0

9 780821 842270